
This paper's fundamental contribution is a novel point of view towards getting exponential concentration inequalities for a random dynamical system, in an extremely tractable manner. To achieve this we connect ideas and techniques from particle methods, functional inequalities and convergence of Markov chains in Wasserstein distance.

For stable LDS, we show how to obtain exponential concentration of the form (REF ) from single trajectory. The idea is that we can find a common metric on the space of probability measures, such that the transition kernels are contractive and uniformly satisfy transportation-entropy inequality.

We show that weighted T-E inequalities, initially developed by [1]} for theoretical exploration, reveal sharp concentration inequality for an empirical average of the test function (evaluated on i.i.d samples from stationary distribution), under the existence assumption of exponential-type Lyapunov function.

In case of non-linear random dynamical systems, such as Harris ergodic Markov chains (HEMCs), we show that if exponential Lyapunov function exists, an exponential concentration of the form is obtained
\( \mathbb {P} \Bigg [ \bigg |\frac{1}{N} \sum _{i=1}^{N} r(y_i) -<r>_{\mu _{\pi }}\bigg | > \epsilon \Bigg ] \le 2\exp \bigg (-\frac{N \epsilon ^2}{K_{sys}(r)}\bigg );\) 
here, \((y_i)_{i \in \mathbb {N}}\)  are i.i.d samples from \(\mu _{\pi }\) . Since, in application we can not simulate i.i.d samples from a stationary distribution of the dynamical system, we can get concentration of the form (REF ) by simulating multiple independent trajectories from (REF ) and averaging the samples after some burn-in time. Reason being, we cannot find a common metric under which the transition kernels are contractive and uniformly satisfy a T-E inequality.Note that we would not even get exponential concentration by running multiple independent trajectories, if not for existence of exponential Lyapunov function

